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Question

Let the vectors $stack a with rightwards arrow on top comma stack b with rightwards arrow on top comma stack c with rightwards arrow on top comma stack d with rightwards arrow on top$ be such that $left parenthesis stack a with rightwards arrow on top cross times stack b with rightwards arrow on top right parenthesis cross times left parenthesis stack c with rightwards arrow on top cross times stack d with rightwards arrow on top right parenthesis equals stack 0 with rightwards arrow on top$ . Let P₁ and P₂ be planes determined by the pairs of vectors $stack a with rightwards arrow on top comma stack b with rightwards arrow on top$ and $stack c with rightwards arrow on top comma stack d with rightwards arrow on top$ respectively. Then, then angle between between P₁ and P₂ is

0
$\frac{π}{4}$
$\frac{π}{3}$
$\frac{π}{2}$

The correct answer is: 0

If $stack n with rightwards arrow on top subscript 1 end subscript$ and $stack n with rightwards arrow on top subscript 2 end subscript$ are normals to planes P₁ and P₂ then $stack n with rightwards arrow on top subscript 1 end subscript$ × $stack n with rightwards arrow on top subscript 2 end subscript$ = 0
 $stack n with rightwards arrow on top subscript 1 end subscript$ | | $stack n with rightwards arrow on top subscript 2 end subscript$  angle between P₁ and P₂ is zero

Related Questions to study

The length of the longest diagonal of the parallelogram constructed on $5 stack a with rightwards arrow on top plus 2 stack b with rightwards arrow on top$ and $stack a with rightwards arrow on top minus 3 stack b with rightwards arrow on top comma$ if it is given that $vertical line stack a with rightwards arrow on top vertical line equals 2 square root of 2 comma vertical line stack b with rightwards arrow on top vertical line equals 3$ and $angle stack a with rightwards arrow on top comma stack b with rightwards arrow on top equals fraction numerator pi over denominator 4 end fraction comma$ is

The length of the longest diagonal of the parallelogram constructed on $5 stack a with rightwards arrow on top plus 2 stack b with rightwards arrow on top$ and $stack a with rightwards arrow on top minus 3 stack b with rightwards arrow on top comma$ if it is given that $vertical line stack a with rightwards arrow on top vertical line equals 2 square root of 2 comma vertical line stack b with rightwards arrow on top vertical line equals 3$ and $angle stack a with rightwards arrow on top comma stack b with rightwards arrow on top equals fraction numerator pi over denominator 4 end fraction comma$ is

ABCD is a quadrilateral with $Error converting from MathML to accessible text.$ and $Error converting from MathML to accessible text.$ . If its area is  times the area of the parallelogram with AB, AD as adjacent sides, then  is

ABCD is a quadrilateral with $Error converting from MathML to accessible text.$ and $Error converting from MathML to accessible text.$ . If its area is  times the area of the parallelogram with AB, AD as adjacent sides, then  is

Let $stack a with rightwards arrow on top comma stack b with rightwards arrow on top$ and $stack c with rightwards arrow on top$ be non-zero vectors such that $left parenthesis stack a with rightwards arrow on top cross times stack b with rightwards arrow on top right parenthesis cross times stack c with rightwards arrow on top equals fraction numerator 1 over denominator 3 end fraction vertical line stack b with rightwards arrow on top vertical line vertical line stack c with rightwards arrow on top vertical line stack a with rightwards arrow on top.$ If  is the acute angle between the vectors $stack b with rightwards arrow on top$ and $stack c with rightwards arrow on top$ , then sin  equals

Let $stack a with rightwards arrow on top comma stack b with rightwards arrow on top$ and $stack c with rightwards arrow on top$ be non-zero vectors such that $left parenthesis stack a with rightwards arrow on top cross times stack b with rightwards arrow on top right parenthesis cross times stack c with rightwards arrow on top equals fraction numerator 1 over denominator 3 end fraction vertical line stack b with rightwards arrow on top vertical line vertical line stack c with rightwards arrow on top vertical line stack a with rightwards arrow on top.$ If  is the acute angle between the vectors $stack b with rightwards arrow on top$ and $stack c with rightwards arrow on top$ , then sin  equals

parallel

If $stack u with rightwards arrow on top comma stack v with rightwards arrow on top$ and $stack w with rightwards arrow on top$ are three non-coplanar vectors, then $left parenthesis stack u with rightwards arrow on top plus stack v with rightwards arrow on top minus stack w with rightwards arrow on top right parenthesis . left parenthesis stack u with rightwards arrow on top minus stack v with rightwards arrow on top right parenthesis cross times left parenthesis stack v with rightwards arrow on top minus stack w with rightwards arrow on top right parenthesis$ equals

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Given two vectors $stack i with hat on top minus stack j with hat on top$ and $stack i with hat on top minus 2 stack j with hat on top$ , then unit vector coplanar with two vectors and perpendicular to first is

Given two vectors $stack i with hat on top minus stack j with hat on top$ and $stack i with hat on top minus 2 stack j with hat on top$ , then unit vector coplanar with two vectors and perpendicular to first is

If vectors $Error converting from MathML to accessible text.$ and $Error converting from MathML to accessible text.$ are the sides of ABC, then the length of the median through A is:

If vectors $Error converting from MathML to accessible text.$ and $Error converting from MathML to accessible text.$ are the sides of ABC, then the length of the median through A is:

parallel

The vectors $stack A B with rightwards arrow on top$ = $3 stack i with hat on top$ + $5 stack j with hat on top$ + $4 stack k with hat on top$ and $stack A C with rightwards arrow on top$ = $5 stack i with hat on top$ – $5 stack j with hat on top$ + $2 stack k with hat on top$ are the sides of a triangle ABC. The length of the median through A is

The vectors $stack A B with rightwards arrow on top$ = $3 stack i with hat on top$ + $5 stack j with hat on top$ + $4 stack k with hat on top$ and $stack A C with rightwards arrow on top$ = $5 stack i with hat on top$ – $5 stack j with hat on top$ + $2 stack k with hat on top$ are the sides of a triangle ABC. The length of the median through A is

Let , ,  be distinct real numbers. The points with position vectors $alpha stack i with hat on top$ + $beta stack j with hat on top$ + $gamma stack k with hat on top$ , $beta stack i with hat on top$ + $gamma stack j with hat on top$ + $alpha stack k with hat on top$ , $gamma stack i with hat on top$ + $alpha stack j with hat on top$ + $beta stack k with hat on top$

Let , ,  be distinct real numbers. The points with position vectors $alpha stack i with hat on top$ + $beta stack j with hat on top$ + $gamma stack k with hat on top$ , $beta stack i with hat on top$ + $gamma stack j with hat on top$ + $alpha stack k with hat on top$ , $gamma stack i with hat on top$ + $alpha stack j with hat on top$ + $beta stack k with hat on top$

ABCD is a parallelogram and P is the intersection point of diagonals. If O is any point in the plane of the parallelogram, then
$stack O A with rightwards arrow on top$ + $stack O B with rightwards arrow on top$ + $stack O C with rightwards arrow on top$ + $stack O D with rightwards arrow on top$ must be equal to -

ABCD is a parallelogram and P is the intersection point of diagonals. If O is any point in the plane of the parallelogram, then
$stack O A with rightwards arrow on top$ + $stack O B with rightwards arrow on top$ + $stack O C with rightwards arrow on top$ + $stack O D with rightwards arrow on top$ must be equal to -

parallel

The non-zero vectors $stack a with rightwards arrow on top$ ¸ $stack b with rightwards arrow on top$ and $stack c with rightwards arrow on top$ are related by $stack a with rightwards arrow on top$ = 8 $stack b with rightwards arrow on top$ and $stack c with rightwards arrow on top$ = – 7 $stack b with rightwards arrow on top$ . Then the angle between $stack a with rightwards arrow on top$ and $stack c with rightwards arrow on top$ is -

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The position vectors of P and Q are respectively $stack a with rightwards arrow on top$ and $stack b with rightwards arrow on top$ . If R is a opint on $stack P Q with left right arrow on top$ such that $stack P R with rightwards arrow on top$ = 5 $stack P Q with rightwards arrow on top$ , then the position vector of R is -

Points A ( $2 stack i with hat on top$ – $stack j with hat on top$ + $stack k with hat on top$ ), B( $stack i with hat on top$ – $3 stack j with hat on top$ – $5 stack k with hat on top$ ), C( $3 stack i with hat on top$ – $4 stack j with hat on top$ – $4 stack k with hat on top$ ) are the vertices of -

Points A ( $2 stack i with hat on top$ – $stack j with hat on top$ + $stack k with hat on top$ ), B( $stack i with hat on top$ – $3 stack j with hat on top$ – $5 stack k with hat on top$ ), C( $3 stack i with hat on top$ – $4 stack j with hat on top$ – $4 stack k with hat on top$ ) are the vertices of -

parallel

If $stack a with rightwards arrow on top. stack i with hat on top equals stack a with rightwards arrow on top. left parenthesis stack i with hat on top plus stack j with hat on top right parenthesis equals a. left parenthesis stack i with hat on top plus stack j with hat on top plus stack k with hat on top right parenthesis$ , then $stack a with rightwards arrow on top$ is equal to-

If $stack a with rightwards arrow on top. stack i with hat on top equals stack a with rightwards arrow on top. left parenthesis stack i with hat on top plus stack j with hat on top right parenthesis equals a. left parenthesis stack i with hat on top plus stack j with hat on top plus stack k with hat on top right parenthesis$ , then $stack a with rightwards arrow on top$ is equal to-

If angle between $stack a with rightwards arrow on top$ and $stack b with rightwards arrow on top$ is obtuse, then

If angle between $stack a with rightwards arrow on top$ and $stack b with rightwards arrow on top$ is obtuse, then

The position vectors of P and Q are respectively $stack a with rightwards arrow on top$ and $stack b with rightwards arrow on top$ . If R is a point on $Error converting from MathML to accessible text.$ such that $Error converting from MathML to accessible text.$ = 5 $Error converting from MathML to accessible text.$ , then the position vector of R is

The position vectors of P and Q are respectively $stack a with rightwards arrow on top$ and $stack b with rightwards arrow on top$ . If R is a point on $Error converting from MathML to accessible text.$ such that $Error converting from MathML to accessible text.$ = 5 $Error converting from MathML to accessible text.$ , then the position vector of R is

parallel

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